1. For the sauce, combine the egg, yolks, vinegar, and mustard in a blender. With the machine running, gradually add the oil to make a thick and creamy mayonnaise. Season with salt and pepper. Transfer to a bowl and stir in the basil.

2. Lightly pound the chicken breasts to an even thickness and season with salt and pepper. Spread the flour in a shallow dish. Mix the bread crumbs, Parmesan cheese, thyme, and parsley in a shallow bowl. Beat the eggs in another bowl. Coat the chicken in the flour, shaking off the excess. Dip in the eggs, then coat with the breadcrumb mixture.

3. Heat the olive and vegetable oils in a large frying pan over medium-high heat until shimmering. Add the chicken and cook, turning once, until golden brown, about 10 minutes. Transfer to paper towels to drain briefly. Serve hot with the sauce passed on the side.